Optimal extensions for p-th power factorable operators

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Let X(μ) be a function space related to a measure space (Ω,Σ,μ) with χΩ∈ X(μ) and let T X(μ)→ E be a Banach space valued operator. It is known that if T is p-th power factorable then the largest function space to which T can be extended preserving p-th power factorability is given by the space Lᵖ(mₜ) of p-integrable functions with respect to mₜ, where mₜΣ→ E is the vector measure associated to T via mₜ(A)=T(χₐ). In this paper we extend this result by removing the restriction χΩ∈ X(μ). In this general case, by considering mₜ defined on a certain δ-ring, we show that the optimal domain for T is the space Lᵖ(mₜ)∩ L¹(mₜ). We apply the obtained results to the particular case when T is a map between sequence spaces defined by an infinite matrix.